It is stated on the Runyon website, as mentioned above - that he played a prototype of the Runyon 22. The trustworthyness of SOTW may be a separate matter.
Here: http://www.saxgourmet.com/runyon/accessory.pdf at the top of page 8, if you are interested.
thanks for that link, as you can see from the URL it's an old Runyon pdf on the Saxgourmet website, Runyon have updated their website and don't have this info anymore. It would seem that the statement "Charlie Parker for a time played a prototype of our Model 22" has been conflated into "Charlie Parker played a Runyon 22" on SOTW.
I can't find out whether the RS Berkeley Charlie Parker Legends series mouthpiece is copied from a Brilhart or a Runyon, it was copied from a white mouthpiece found in the case of a King Super 20 belonging to Kim Parker, so presumably it dates from the 50's when Bird was playing a King sax.
but digression aside, to return to the original question....
If you're gonna play jazz you've got to develop your embouchure and have enough to say about life that you can improvise at length without just running through a load of pre learned licks and phrases. Knowing all the standards will also help, since they're the basic tunes that people play at jam sessions, and going out and jamming with everyone you can find is part of the process.
No mouthpiece will give you an instant jazz sound, but the one's recommended above will at least play well, so you won't be struggling against a substandard mouthpiece. If I had to play jazz tomorrow, I'd get out my Jody Jazz HR and be confident that it'd do the job. The Phil-Tone mouthpieces have a great reputation and would be top of my shopping list if I didn't already have the Jody. There are a lot of 'improved Meyer' type alto mouthpieces around, it's become the orthodoxy that you should play a Meyer on alto, but the modern Meyers have a reputation for poor quality control, so a number of mouthpiece makers have started doing their own improved versions of these pieces.
Get a good mouthpiece and learn how to use it. Musical instrument forums are full of people who're constantly changing their equipment and one wonders how much actual music they play. I've been as guilty of that as anyone and have a few boxes of mouthpieces to prove it. It's fun trying different mouthpieces and saxes etc, but it's also a distraction from the real task of being a musician and creating music. Go out and try different mouthpieces until you find the one you like the most and then get on with playing in a band. There's nothing better than playing a solo with a good rhythm section behind you. Even if you can't think of anything to play and are struggling, it's still a learning experience and sometimes what comes out of your horn in such testing circumstances is better than if you're not struggling - the extra effort brings out something more...
Using a stronger reed than a 1 1/2 might be a good idea, there are a few pro players who use soft reeds, often because they're playing mouthpieces with very wide tip openings, but in general, it'll help if you work your way up to a 2, 2 1/2 or 3.
While everyone will recommend their own favourite make of reed, I've tried to avoid getting hung up on any particular make because when you're out on the road you can sometimes find that you can't get hold of your favourite and have to make do with whatever you can get at the time. I like Vandoren Javas for their flexibility, other people don't like them at all...
Bear in mind that reed strengths vary from different manufacturers and Vandorens are about a quarter strength harder than Ricos and there's always some variation - no two reeds are quite the same
there's a comparison chart here which shows the differences in reed strengths from different companies -
http://www.sax.co.uk/rcc.htm